Higher order Fourier analysis and algebraic property testing Hamed - - PowerPoint PPT Presentation

Higher order Fourier analysis and algebraic property testing

Hamed Hatami

School of Computer Science McGill University

July 22, 2019

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 1 / 26

Based on joint works with Arnab Bhattacharyya, Eldar Fischer, Pooya Hatami, Shachar Lovett.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 2 / 26

Property Testing

Given a function:

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

Property Testing

Given a function: Evaluate it on a small number of points:

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

Property Testing

Given a function: Evaluate it on a small number of points: Decide whether

◮ it satisfies a property P ◮ or is “far” from satisfying P.

P far from P

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

Property Testing

Given a function: Evaluate it on a small number of points: Decide whether

◮ it satisfies a property P ◮ or is “far” from satisfying P.

P far from P

Definition

dist(f, g) = Pr[f(x) = g(x)]. dist(f, P) = ming∈P dist(f, g).

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Closely related to the concepts of regularity and uniformity [Ruzsa-Szemerédi 76], [Rödl-Duke 85].

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Closely related to the concepts of regularity and uniformity [Ruzsa-Szemerédi 76], [Rödl-Duke 85]. Formally defined by [Rubinfeld, Sudan 96], [Goldreich, Goldwasser, Rubinfeld 98].

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Closely related to the concepts of regularity and uniformity [Ruzsa-Szemerédi 76], [Rödl-Duke 85]. Formally defined by [Rubinfeld, Sudan 96], [Goldreich, Goldwasser, Rubinfeld 98]. Closely related to limit theories of combinatorial objects [Lovász-Szegedy 2010].

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

Example

Let P = {functions f : Fn

p → {0, 1} where f ≡ 0}.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 5 / 26

Example

Let P = {functions f : Fn

p → {0, 1} where f ≡ 0}.

Test

Pick x ∈ Fn

p at random.

If f(x) = 0 accept

• therwise reject.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 5 / 26

Example

Let P = {functions f : Fn

p → {0, 1} where f ≡ 0}.

Test

Pick x ∈ Fn

p at random.

If f(x) = 0 accept

• therwise reject.

Analysis

If f ∈ P, then Pr[accept] = 1. If dist(f, P) > ǫ, then Pr[reject] ≥ ǫ.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 5 / 26

Property Testing

In this talk we focus on the following version of property testing: One-sided error: If f ∈ P, then Pr[accept] = 1.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 6 / 26

Property Testing

In this talk we focus on the following version of property testing: One-sided error: If f ∈ P, then Pr[accept] = 1. If dist(f, P) ≥ ǫ > 0, then Pr[reject] ≥ δ(ǫ) > 0.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 6 / 26

Property Testing

In this talk we focus on the following version of property testing: One-sided error: If f ∈ P, then Pr[accept] = 1. If dist(f, P) ≥ ǫ > 0, then Pr[reject] ≥ δ(ǫ) > 0. Proximity Oblivious: Number of queries is a fixed constant that does not depend on ǫ.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 6 / 26

BLR linearity testing

Linearity: f : Fn

2 → F2, f(x) = a1x1 + . . . + anxn.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing

Linearity: f : Fn

2 → F2, f(x) = a1x1 + . . . + anxn.

Local characterization: f(x + y) = f(x) + f(y) for all x, y.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing

Linearity: f : Fn

2 → F2, f(x) = a1x1 + . . . + anxn.

Local characterization: f(x + y) = f(x) + f(y) for all x, y.

Test

Pick x, y ∈ Fn

2 at random.

If f(x + y) = f(x) + f(y) accept

• therwise reject.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing

Linearity: f : Fn

2 → F2, f(x) = a1x1 + . . . + anxn.

Local characterization: f(x + y) = f(x) + f(y) for all x, y.

Test

Pick x, y ∈ Fn

2 at random.

If f(x + y) = f(x) + f(y) accept

• therwise reject.

Analysis

If f linear, then Pr[accept] = 1.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing

Linearity: f : Fn

2 → F2, f(x) = a1x1 + . . . + anxn.

Local characterization: f(x + y) = f(x) + f(y) for all x, y.

Test

Pick x, y ∈ Fn

2 at random.

If f(x + y) = f(x) + f(y) accept

• therwise reject.

Analysis

If f linear, then Pr[accept] = 1. dist(f, P) > ǫ = ⇒ max f(S) < 1 − 2ǫ = ⇒ Pr[reject] ≥ ǫ.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

Classical Fourier Analysis over Fn

2

For a ∈ Fn

2, define χa : x → (−1)a1x1+...+anxn.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 8 / 26

Classical Fourier Analysis over Fn

2

For a ∈ Fn

2, define χa : x → (−1)a1x1+...+anxn.

These characters are orthonormal: χa, χb = Exχa(x)χb(x) = 1 a = b a = b .

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 8 / 26

Classical Fourier Analysis over Fn

2

For a ∈ Fn

2, define χa : x → (−1)a1x1+...+anxn.

These characters are orthonormal: χa, χb = Exχa(x)χb(x) = 1 a = b a = b . Every f : Fn

2 → R is uniquely expanded as

f =

• a
• f(a)χa.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 8 / 26

Linearity and Fourier Analysis

f : Fn

2 → F2.

Change the range to f : Fn

2 → {−1, 1} (by considering (−1)f).

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Linearity and Fourier Analysis

f : Fn

2 → F2.

Change the range to f : Fn

2 → {−1, 1} (by considering (−1)f).

f : Fn

2 → {−1, 1} linear ⇔ it is a character f = χa.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Linearity and Fourier Analysis

f : Fn

2 → F2.

Change the range to f : Fn

2 → {−1, 1} (by considering (−1)f).

f : Fn

2 → {−1, 1} linear ⇔ it is a character f = χa.

If dist(f, Linear) > ǫ then f is far from all characters: max f(S) < 1 − 2ǫ.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Linearity and Fourier Analysis

f : Fn

2 → F2.

Change the range to f : Fn

2 → {−1, 1} (by considering (−1)f).

f : Fn

2 → {−1, 1} linear ⇔ it is a character f = χa.

If dist(f, Linear) > ǫ then f is far from all characters: max f(S) < 1 − 2ǫ. BLR: Then Pr[reject] = 1 2(1 − Ef(x)f(y)f(x + y)) ≥ ǫ.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Algebraic Property Testing

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 10 / 26

Our setting

Functions of the form f : Fn

p → {0, . . . , R} where

p is a fixed prime. R is a fixed integer.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 11 / 26

Our setting

Functions of the form f : Fn

p → {0, . . . , R} where

p is a fixed prime. R is a fixed integer. Two important cases: R = 1: Functions f : Fn

p → {0, 1}.

R = p − 1: Functions f : Fn

p → Fp.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 11 / 26

Our setting

Functions of the form f : Fn

p → {0, . . . , R} where

p is a fixed prime. R is a fixed integer. Two important cases: R = 1: Functions f : Fn

p → {0, 1}.

R = p − 1: Functions f : Fn

p → Fp.

What conditions should we impose on P?

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 11 / 26

Our setting

Functions of the form f : Fn

p → {0, . . . , R} where

p is a fixed prime. R is a fixed integer. Two important cases: R = 1: Functions f : Fn

p → {0, 1}.

R = p − 1: Functions f : Fn

p → Fp.

What conditions should we impose on P? We do not want to treat Fn

p as a generic set of size pn and ignore

the algebraic structure of Fn

p.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 11 / 26

Note

Symmetry plays an important role in property testing. E.g. graph properties are invariant under permutations of vertices.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 12 / 26

Note

Symmetry plays an important role in property testing. E.g. graph properties are invariant under permutations of vertices.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 12 / 26

Kaufman-Sudan

P is called affine-invariant if it is invariant under any affine transformation: f ∈ P ⇒ f ◦ A ∈ P for any affine transformation A : Fn

p → Fn

• p. (i.e. A : x → Bx + c)

Note

Symmetry plays an important role in property testing. E.g. graph properties are invariant under permutations of vertices.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 12 / 26

Kaufman-Sudan

P is called affine-invariant if it is invariant under any affine transformation: f ∈ P ⇒ f ◦ A ∈ P for any affine transformation A : Fn

p → Fn

• p. (i.e. A : x → Bx + c)

Example

P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Note

Symmetry plays an important role in property testing. E.g. graph properties are invariant under permutations of vertices.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 12 / 26

Question

Which affine-invariant properties P are testable?

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 13 / 26

Question

Which affine-invariant properties P are testable?

Example

P = {Linear f : Fn

p → Fp}.

Local characterization: f(x + y) = f(x) + f(y).

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 13 / 26

Example

P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 14 / 26

Example

P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Local Characterization of P

f ∈ P ⇐ ⇒ f|V ∈ P for all affine subspace V ⊆ Fn

p with dim(V) = d + 1.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 14 / 26

Example

P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Local Characterization of P

f ∈ P ⇐ ⇒ f|V ∈ P for all affine subspace V ⊆ Fn

p with dim(V) = d + 1.

Test for deg ≤ d.

Pick a d + 1-dimensional random affine subspace V ⊆ Fn

p.

Accept if deg(f|V) ≤ d, and reject otherwise.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 14 / 26

Example

P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Local Characterization of P

f ∈ P ⇐ ⇒ f|V ∈ P for all affine subspace V ⊆ Fn

p with dim(V) = d + 1.

Test for deg ≤ d.

Pick a d + 1-dimensional random affine subspace V ⊆ Fn

p.

Accept if deg(f|V) ≤ d, and reject otherwise. if f ∈ P then Pr[accept] = 1.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 14 / 26

Example

P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Local Characterization of P

f ∈ P ⇐ ⇒ f|V ∈ P for all affine subspace V ⊆ Fn

p with dim(V) = d + 1.

Test for deg ≤ d.

Pick a d + 1-dimensional random affine subspace V ⊆ Fn

p.

Accept if deg(f|V) ≤ d, and reject otherwise. if f ∈ P then Pr[accept] = 1. if dist(f, P) ≥ ǫ then Pr[reject] > δ(ǫ) > 0. [Alon, Kaufman, Krivelevich, Litsyn, Ron 2005].

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 14 / 26

Local characterization

P is locally characterizable if there exists k > 0 such that f ∈ P ⇐ ⇒ f|V ∈ P for all affine subspace V ⊆ Fn

p with dim(V) = k.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 15 / 26

Local characterization

P is locally characterizable if there exists k > 0 such that f ∈ P ⇐ ⇒ f|V ∈ P for all affine subspace V ⊆ Fn

p with dim(V) = k.

Theorem (Bhattacharyya, Fischer, HH, P . Hatami, and Lovett)

Every locally characterizable property is testable.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 15 / 26

Approach: Higher-order Fourier Analysis

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 16 / 26

Classical Fourier analysis

Consider f : Fn

2 → {−1, 1}.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 17 / 26

Classical Fourier analysis

Consider f : Fn

2 → {−1, 1}.

Let S = {a : | f(a)| ≥ δ} all significant Fourier coefficients.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 17 / 26

Classical Fourier analysis

Consider f : Fn

2 → {−1, 1}.

Let S = {a : | f(a)| ≥ δ} all significant Fourier coefficients. f =

• a∈S
• f(a)χa +
• b∈S
• f(b)χb = f1 + f2 ≈ f1.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 17 / 26

Classical Fourier analysis

Consider f : Fn

2 → {−1, 1}.

Let S = {a : | f(a)| ≥ δ} all significant Fourier coefficients. f =

• a∈S
• f(a)χa +
• b∈S
• f(b)χb = f1 + f2 ≈ f1.

f1 is the linearly structured part.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 17 / 26

Classical Fourier analysis

Consider f : Fn

2 → {−1, 1}.

Let S = {a : | f(a)| ≥ δ} all significant Fourier coefficients. f =

• a∈S
• f(a)χa +
• b∈S
• f(b)χb = f1 + f2 ≈ f1.

f1 is the linearly structured part. f2 is noise.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 17 / 26

Quadratic functions f : Fn

2 → F2

Consider the quadratic function f(x) = (−1)x1x2+x3x4+...+xn−1xn .

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 18 / 26

Quadratic functions f : Fn

2 → F2

Consider the quadratic function f(x) = (−1)x1x2+x3x4+...+xn−1xn . All the Fourier coefficients of f are very small.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 18 / 26

Quadratic functions f : Fn

2 → F2

Consider the quadratic function f(x) = (−1)x1x2+x3x4+...+xn−1xn . All the Fourier coefficients of f are very small. In other words, f have very small correlation with every character.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 18 / 26

Quadratic functions f : Fn

2 → F2

Consider the quadratic function f(x) = (−1)x1x2+x3x4+...+xn−1xn . All the Fourier coefficients of f are very small. In other words, f have very small correlation with every character. Fourier analysis cannot detect the quadratic structure of f. It is pure noise from its perspective.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 18 / 26

Quadratic functions f : Fn

2 → F2

Consider the quadratic function f(x) = (−1)x1x2+x3x4+...+xn−1xn . All the Fourier coefficients of f are very small. In other words, f have very small correlation with every character. Fourier analysis cannot detect the quadratic structure of f. It is pure noise from its perspective. A classical Fourier analytic approach does not distinguish this function from a random function, and cannot analyze a proper test.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 18 / 26

Quadratic Fourier Anlysis

In Quadratic Fourier analysis f ≈ k

i=1 ak × (−1)Qi, where Qi’s

are quadratic polynomials.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 19 / 26

Quadratic Fourier Anlysis

In Quadratic Fourier analysis f ≈ k

i=1 ak × (−1)Qi, where Qi’s

are quadratic polynomials. We think of (−1)Qi as quadratic characters.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 19 / 26

Quadratic Fourier Anlysis

In Quadratic Fourier analysis f ≈ k

i=1 ak × (−1)Qi, where Qi’s

are quadratic polynomials. We think of (−1)Qi as quadratic characters. They are not orthonormal, but they can be made almost

• rthonormal through the notion of rank.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 19 / 26

Quadratic Fourier Anlysis

In Quadratic Fourier analysis f ≈ k

i=1 ak × (−1)Qi, where Qi’s

are quadratic polynomials. We think of (−1)Qi as quadratic characters. They are not orthonormal, but they can be made almost

• rthonormal through the notion of rank.

To apply these, one writes f = f1 + f2, where f2 is small noise, and f1 = c

i=1 ac × (−1)Qi, for a high rank set of quadratics.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 19 / 26

Examples of locally characterizable properties

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 20 / 26

Example

Testable P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 21 / 26

Example

Testable P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Definition (Degree structural properties)

Fix d1, . . . , dc and Γ : Fc

p → {0, 1, . . . , R}.

The property of being expressible as Γ(P1, . . . , Pc) where deg(Pi) ≤ di.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 21 / 26

Example

Testable P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Definition (Degree structural properties)

Fix d1, . . . , dc and Γ : Fc

p → {0, 1, . . . , R}.

The property of being expressible as Γ(P1, . . . , Pc) where deg(Pi) ≤ di.

Example

Polynomials f : Fn

p → Fp that are products of two quadratics.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 21 / 26

Example

Testable P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Definition (Degree structural properties)

Fix d1, . . . , dc and Γ : Fc

p → {0, 1, . . . , R}.

The property of being expressible as Γ(P1, . . . , Pc) where deg(Pi) ≤ di.

Example

Polynomials f : Fn

p → Fp that are products of two quadratics.

Polynomials f : Fn

p → Fp that are squares of a quadratics.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 21 / 26

Example

Testable P = {Polynomials f : Fn

p → Fp of degree ≤ d}.

Definition (Degree structural properties)

Fix d1, . . . , dc and Γ : Fc

p → {0, 1, . . . , R}.

The property of being expressible as Γ(P1, . . . , Pc) where deg(Pi) ≤ di.

Example

Polynomials f : Fn

p → Fp that are products of two quadratics.

Polynomials f : Fn

p → Fp that are squares of a quadratics.

Polynomials f : Fn

p → Fp of the form f = ab + cd where a, b, c, d

are cubics.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 21 / 26

Theorem (Bhattacharyya, Fischer, HH, P . Hatami, and Lovett)

Every degree structural property is locally characterizable and hence testable.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 22 / 26

Theorem (Bhattacharyya, Fischer, HH, P . Hatami, and Lovett)

Every degree structural property is locally characterizable and hence testable. Our proof uses higher order Fourier analysis.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 22 / 26

Theorem (Bhattacharyya, Fischer, HH, P . Hatami, and Lovett)

Every degree structural property is locally characterizable and hence testable. Our proof uses higher order Fourier analysis. Consequently does not provide any reasonable bound on the dimension.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 22 / 26

Open Problems

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 23 / 26

Find a direct proof (with reasonable bounds) for the fact that degree-structural properties are locally characterizable.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 24 / 26

Find a direct proof (with reasonable bounds) for the fact that degree-structural properties are locally characterizable. Is there a good way to estimate the locality dimension of a degree-structural property?

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 24 / 26

Find a direct proof (with reasonable bounds) for the fact that degree-structural properties are locally characterizable. Is there a good way to estimate the locality dimension of a degree-structural property?

Example

What are the locality dimensions? Polynomials f : Fn

p → Fp that are products of two quadratics.

Polynomials f : Fn

p → Fp that are squares of a quadratics.

Polynomials f : Fn

p → Fp of the form f = ab + cd where a, b, c, d

are cubics.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 24 / 26

Open Problems: Beyond locally characterizable

Definition

An affine-invariant property P is affine subspace hereditary if the restriction of any f|V ∈ P for any affine subspace V ⊆ Fn

p.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 25 / 26

Open Problems: Beyond locally characterizable

Definition

An affine-invariant property P is affine subspace hereditary if the restriction of any f|V ∈ P for any affine subspace V ⊆ Fn

p.

Conjecture [Bhattacharyya,Grigorescu,Shapira 2010]

Every affine subspace hereditary property is testable with one-sided error.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 25 / 26

Open Problems: Beyond locally characterizable

Definition

An affine-invariant property P is affine subspace hereditary if the restriction of any f|V ∈ P for any affine subspace V ⊆ Fn

p.

Conjecture [Bhattacharyya,Grigorescu,Shapira 2010]

Every affine subspace hereditary property is testable with one-sided error.

Theorem (Bhattacharyya, Fischer, HH, P . Hatami, and Lovett)

Every affine subspace hereditary property of “bounded complexity” is testable with one-sided error.

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 25 / 26

Thank you!

Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 26 / 26